0xf504a0e4…3f84sent to0xf504a0e4…3f84·#22,554,395·view on Etherscan
\title{\textbf{Probability Tokenization} \\ A versatile approach for automatic prediction market}
\author[1]{\textbf{catslovefish.eth}}
\author[1]{\textbf{Edward Lee}}
\author[1]{\textbf{Arwen Yi}}
\affil[1]{Bayes Labs Research Team}
\pgfplotsset{compat=1.18}
\begin{document}
\maketitle
\renewcommand{\abstractname}{}
\begin{abstract}
\noindent \textbf{Abstract.}We present a probability-tokenization scheme in which each outcome of a random phenomenon is represented by an ERC-20-compatible token, fully backed by USD through one bonding curve.
\begin{itemize}
\item \emph{Tokenization.} Define a mapping
\[
\tau:\,\underbrace{\Omega}_{\text{sample space}}
\longrightarrow
\underbrace{S\subseteq\mathbb R^{N}}_{\text{vector space}},
\qquad
\tau(\operatorname{outcome}_i)=s_i.
\]
Here \(\Omega=\{\operatorname{outcome}_1,\dots,\operatorname{outcome}_N\}\) and
\(
S=\{(s_1,\dots,s_N)\mid s_i\in\mathbb R\}\subseteq\mathbb R^{N},
\)
the token-supply vectors.
\item \emph{Reserve–Token-set bonding.}
We restrict the ambient space $\mathbb R^{N}$ to its \emph{one–dimensional diagonal subspace}
\[
D \;:=\; \operatorname{span}\{\mathbf 1_{N}\}
\;=\;\{(x,\ldots,x)\mid x\in\mathbb R\}.
\]
The linear isomorphism
\[
\hat T:\mathbb R \;\longrightarrow\; D,
\qquad
\hat T(x)=x\,\mathbf 1_{N},
\]
sends a scalar reserve level $x$ to the supply vector
$\hat T(x)$, which always lies on that diagonal.
\end{itemize}
Thus, each USD deposit mints—or each withdrawal burns—an equal quantity of every outcome token, rigidly coupling the total token supply to the collateral pool. This coupling enforces a conservation law on two levels:
\begin{enumerate}[leftmargin=1.8em]
\item \emph{Global / fraction perspective} — Because the $N$ mutually exclusive outcomes partition the sample space, their probabilities satisfy $\sum_{i=1}^{N} p_i = 1$.
Define the \emph{normalised supply fractions}
\[
f_i \coloneqq \frac{s_i}{\sum_{j=1}^{N} s_j}=\frac{1}{N},\quad i=1, \dots,N
\;\;\Longrightarrow\;\;
\sum_{i=1}^{N} f_i = 1.
\] These fractions therefore coincide with the probabilities obtained from the boundary perspective.
\item \emph{Local / derivative perspective} —
Each trade can be decomposed via partial derivatives, showing the receiving of “a bit more of the token $s_i$ ” in place of “a bit less tokens in \( \mathbb D \setminus \{s_i\} \)” . Our key design choice is to restrict $\sum _{i}^N p_{i, USD} = 1$ and thereby enforcing a "probability-like" partition of value across the N outcomes.
\end{enumerate}
\noindent
This scalable, versatile approach creates a vibrant marketplace—a playground where dispersed information is traded, prices encode the crowd’s evolving belief about each event. Over time, these prices either converge to the event’s true probability (from a Frequentist perspective) or progressively update the posterior probability $P(World | Outcome j)$ (from a Bayesian perspective).
\end{abstract}
\section*{0}
\subsection*{0.0 \ In the Language of Category Theory}
For the reader who wish view our paper from a pure mathematical perspective, here is the insight: \footnote{Loop \href{https://www.youtube.com/watch?v=RqHbD1i4NF4}{Glass Animals – “Creatures in Heaven”} while reading, you might feel as I feel — looking for a structure-preserving mapping...}
\[
S : (\Omega,\mathcal{F},\mathbb{P}) \;\longrightarrow\; (T^{\ast}M,\mathcal{B})
\]
where: \begin{itemize}
\item $(\Omega,\mathcal{F},\mathbb{P})$ is a probability triple, ruled by the $\sigma$-algebra $\mathcal{F}$;
\item $T^{\ast}M$ is the cotangent bundle of a smooth (Riemannian) manifold $M$—the natural “price space” when each price is a covector acting linearly on supply vectors; $\mathcal{B}$ is its Borel $\sigma$-algebra.
\end{itemize}
\subsection*{0.1\ Overall Diagram}
\begin{tikzpicture}[scale=0.75,transform shape, % ← scales everything
block/.style={
draw,
rectangle,
rounded corners,
minimum width=3cm,
minimum height=1cm,
align=center
},
>=Stealth,
node distance=3.5cm
]
% First row
\node[block] (A1) {Sample Space};
\node[block,right=of A1] (A2) {
Probability Space};
\node[block,right=of A2] (A3) {\( \{ p_1',\ldots,p'_N \} \)};
\node[block,right=of A3] (A4) {A world with \\ Hypothesis j Prevails };
% Second row
\node[block,below=of A1] (B1) {Vector Space };
\node[block,below=of A2] (B2) {One–dimensional \\ diagonal subspace};
\node[block,below=of A3] (B3) {\( ( q_1',\ldots,q'_N ) \) \\=
\( ( x',\ldots,x' ) \)};
\node[block,below=of A4] (B4) {Result Settlement};
% Third row
\node[block,below=of B1] (C1) {Covariant Vector Space};
\node[block,below=of B2] (C2) {Price Space};
\node[block,below=of B3] (C3) {{\( ( p_{1,u}',\ldots,p'_{N,u} ) \)}};
\node[block,below=of B4] (C4) {$p_{j,u}=1$};
%-------- Row Edges with labels-------
\draw[->] (A1) -- node[midway, above]{\emph{Measurement}}
node[midway, below]{\emph{\(\sum_{i=1}^{N}p_i = 1\)}}(A2);
\draw[->] (A2) -- node[midway, above]{\emph{Time Flows}} (A3);
\draw[->] (A3) -- node[midway, above]{\emph{Time Flows $\ldots$}} (A4);
\draw[->] (B1) -- node[midway, above]{\emph{Bonding Curve}}
node[midway, below]{\emph{Subspace}}(B2);
\draw[->] (B2) --
node[midway, below]{\emph{Info Exchange $\ldots$}}
node[midway, above]{\emph{Token Swap}} (B3);
\draw[->] (B3) --
node[midway, below]{\emph{Info Exchange $\ldots$}}
node[midway, above]{\emph{Token Swap}}
(B4);
\draw[->] (C1) -- node[midway, above]{\(\sum_{i=1}^{N}p_{i,u} = 1\)} (C2);
\draw[->] (C2) --
node[midway, above]{\emph{Price Update}} (C3);
\draw[->] (C3) --
node[midway, above]{\emph{Price Update}}
(C4);
%-------- Column Edges with labels-----
\draw[->] (A1) -- node[midway, above]{\emph{Tokenization}} (B1);
\draw[<->] (A2) -- node[midway, above]{\emph{Price–Probability Duality}} (B2);
\draw[->] (B3) -- node[midway, above]{\emph{Crowd Belief Updated}} (A3);
\draw[->] (B4) -- node[midway, above]{\emph{We believe with strong confidence}} (A4);
\draw[<->] (B1) -- node[midway, above]{\emph{Dual}} (C1);
\draw[<->] (B2) -- node[midway, above]{\emph{Dual}} (C2);
\draw[<->] (B3) -- node[midway, above]{\emph{Dual}} (C3);
\draw[<->] (B4) -- node[midway, above]{\emph{Dual}} (C4);
\end{tikzpicture}
\subsection*{0.2\ Glossary of Symbols}
\begin{table}[h]
\centering
\renewcommand{\arraystretch}{1.2}
\begin{tabular}{@{}llp{8.2cm}@{}}
\textbf{Symbol} & \textbf{Type / space} & \textbf{Meaning} \\ \hline
$\Omega$ & set & Sample space of the random experiment \\
${outcome}_i$ & element of $\Omega$ & $i^{\text{th}}$ atomic outcome \\
$S=\{(s_1,\dots,s_N)\mid s_i\in\mathbb R\}$,
& vector in $\mathbb R^{N}$ & ERC-20 token identifiers (one per outcome) \\
$s_i$ & element of $S$ & Token supply that corresponds to $\operatorname{outcome}_i$ \\
$D=\operatorname{span}\{\mathbf 1_N\}$ & subspace of $\mathbb R^{N}$ & One-dimensional diagonal subspace \\
$\mathbf 1_N$ & vector in $D$ & All-ones column vector $(1,\dots,1)^{\top}$ \\
$\hat T$ & linear map & Isomorphism $\hat T:\mathbb R\!\to\!D,\;\hat T(x)=x\mathbf 1_N$ \\
$x$ & real number & Amount of collateral in USD \\
$\hat T(x)$ & vector in $D$ & Current token-supply vector \\
$p_i$ & real number & Probability weight assigned to outcome $i$ \\
$p_{i, USD}$ & real number & Marginal price $\displaystyle\frac{\partial(\text{USD})}{\partial q_i}$ \\
$p_{XY} = \frac{\partial Y}{\partial X}$ & real number & Relative price \( p_{XY} \) of token \( X \) in terms of token \( Y \)\tablefootnote{This represents the amount of token \( Y \) required to mint an additional unit of token \( X \). Additionally, it should be noted that \( p_{YX} = \frac{\partial X}{\partial Y} = \frac{1}{p_{XY}} \).}\\
Y, N, D & &fungible tokens representing the yes, no and draw outcome \\
$\{s_Y,s_N,s_D\}$ & &quantity of token set supplied by bonding curve\\
$f: \mathbb{R}^N \rightarrow \mathbb{R} $ & &mapping (function) from a set to a number \\
$r_U$ & & quantity of token USD\tablefootnote{In principle, this base token can be any other fungible token like DAI, ETH or SOL. We use USD here for
reading fluency. } deposited (locked) into the bonding curve contract \\
\end{tabular}
\caption{Glossary of symbols}
\end{table}
\medskip
\noindent\textit{Notation note.}
Throughout the paper we occasionally write the token label \(s_i\) and its
current quantity \(x\) in a context-dependent way.
Strictly, \(s_i\in S\) is a \emph{key} (the \(i\)-th ERC-20 identifier), whereas
\(x\in\mathbb R_{\ge0}\) is the corresponding \emph{value} (the amount of that
token in circulation). Whenever the subscript \(i\) makes the pairing
unambiguous, we treat the two symbols interchangeably; e.g.\ “mint one more
\(s_i\)” is shorthand for “increase \(x\) by one unit.”
\section{Introduction}
\begin{tcolorbox}[title= "Why Prediction Matters?", colback=gray!0,
colframe=gray!60!black]
Humans are thinking reeds—powerful in will and reason yet fragile before the cosmos. From the moment our ancestors dropped from the trees to forage and hunt, \textbf{prediction} has been a matter of survival. Whether choosing when to harvest, where to invest, or how to navigate conflict, anticipating tomorrow undergirds nearly every endeavor. Ancient societies sought guidance from oracles and shamans; today we turn to increasingly sophisticated models. Yet even the fastest supercomputers cannot capture every variable of multi‑body physics or quantum‑scale biology, and expert forecasters still miss the mark with unsettling frequency.
\end{tcolorbox}
\begin{tcolorbox}[title="From Oracles to Market: A Brief History", colback=gray!0,
colframe=gray!60!black]
Wagering on the future is hardly new. Records from the early 1500s show bets on papal succession, and by the late 19th century Wall-Street hosted vibrant U.S. election pools. In 1907 Francis Galton’s county‑fair ox‑weighing exercise revealed that a crowd’s median estimate could outshine individual experts, foreshadowing the “wisdom of crowds.” Twentieth‑century economists such as Friedrich Hayek and Ludwig von Mises laid the theoretical groundwork by viewing prices as information signals.
\smallskip
Modern attempts to formalize crowd forecasting gained momentum with Robin Hanson’s advocacy and the Iowa Electronic Markets, first deployed during the 1988 U.S. presidential campaign. The internet has since given birth to public platforms—Kalshi, Polymarket, Manifold, and others—that turn forecasts into tradable assets.
\end{tcolorbox}
\begin{tcolorbox}[title="Prediction Markets: Playground for Crowd Belief", colback=gray!0,
colframe=gray!60!black]
Prediction markets operationalize Hayek’s insight: when traders back their beliefs with real stakes, prices converge to probabilistic forecasts that often rival—or surpass—experts and algorithms. Carefully designed incentive mechanisms align self‑interest with information revelation, making markets a practical instantiation of \textit{game‑theoretic} principles. Empirical studies, field deployments, and commercial performance all attest to their accuracy.
\end{tcolorbox}
\begin{tcolorbox}[title="Liquidity: The Versatile Challenge",colback=gray!0,
colframe=gray!60!black]
If prediction markets are so powerful, why are they still a niche? Their Achilles’ heel is \textit{liquidity}. Market makers shoulder tail risk—ending up with worthless shares if the event breaks against them—which discourages sustained depth without subsidies. Thin order books impair price discovery, while illiquidity on niche or long‑tail events limits the very diversity that makes markets valuable. Overcoming this paradox—through better mechanisms, regulation, or pooled liquidity—remains the central challenge to scaling crowd forecasting into the mainstream.
\end{tcolorbox}
\begin{tcolorbox}[title= "How we approach this challenge", colback=gray!0,
colframe=gray!60!black]
This paper proposes how \textbf{\textit{Bonding curve-based Automated Prediction Market Maker (BAPMM)}}—can empower new on-chain prediction markets with continuous liquidity management, automatic result settlement. Our objective is to blend a wide spectrum of crowd belief with solid incentive structures, yielding systems that surpass conventional order books in transparency, scalability, and composability. The rest of this paper proceeds as outlined below.
\begin{itemize} \item \textbf{Section 2: Review of Existing Mechanisms
}. \item \textbf{Section 3: The Proposed BAPMM} -- what is it, and how it is derived. \item \textbf{Section 4: Open Questions} -- Discusses potential challenges and future directions. \item \textbf{Section 5: Acknowledgments} \end{itemize}
\end{tcolorbox}
% ------------------------------------------------------------
\section{Review of Existing Mechanisms}\label{sec:mechanism-review}
\begin{tcolorbox}[title={Evaluation Metrics for Prediction Maket},
colback=gray!5, colframe=gray!65!black,
boxrule=0.6pt, arc=3pt,
left=6pt,right=6pt,top=4pt,bottom=4pt]
\begin{enumerate}
\item \textbf{Protocol Versatility} – headline \emph{and} long-tail events remain tradable irrespective of participant count.
\item \textbf{System Acquisition Capability} – market prices aggregate and reflect collective beliefs.
\item \textbf{Price Sum-to-One} – quoted outcome prices must always total 1.
\item \textbf{Smooth Settlement} – winning outcome deterministically settles at price = 1 without manual steps.
\end{enumerate}
\end{tcolorbox}
\noindent
With these yardsticks in place, we now test the three dominant designs
found in prediction markets:
\subsection{Traditional Bookmaker}
% ----------- boxed “Mechanism Overview” -----------------
\begin{tcolorbox}[
title={Mechanism Overview},
enhanced,
colback=white,
colframe=blue!80!black,
colbacktitle=blue!10,
fonttitle=\bfseries\sffamily,
coltitle=black,
boxrule=0.6pt, arc=2pt,
left=8pt,right=6pt,top=4pt,bottom=4pt]
Traditional bookmakers operate as centralized counterparties that
publish “static, pre-event odds’’ derived from historical data and
proprietary risk models.
When a bettor places a wager, the bookmaker immediately takes the
opposite side, updating its internal liability book but \emph{not} the
publicly quoted odds. In effect, the bookmaker is the sole liquidity
provider and price setter.
\end{tcolorbox}
\subsubsection*{Illustrative Example}
\begin{center}
\small % -- overall down-scale --------------------------------
% -------- LEFT COLUMN -------------------------------------------------
\begin{minipage}[t]{0.49\textwidth}\vspace{0pt}
\begin{enumerate}[leftmargin=*, itemsep=3pt, topsep=0pt]
\item \textbf{Odds Posting.}
The bookmaker lists a match with fixed odds of 2.00 (50\%) on Team~A and
1.80 (55.6\%) on Team~B, embedding an over-round of 5.6\%.
\item \textbf{User Bet Placement.}
A bettor logs in, selects Team~A, and stakes \$10\,000.
The interface confirms a potential payout of \$20\,000 at the posted odds.
\item \textbf{Instant Acceptance.}
Because the bookmaker is the counter-party, the wager is accepted immediately;
liability on Team~A now stands at \$10\,000.
\item \textbf{Market Stagnation.}
Unless the bookmaker manually reposts the line or hedges elsewhere, the public odds
shown to the next bettor remain 2.00, even though one large bet has shifted sentiment.
\item \textbf{Settlement.}
After the match ends, the bookmaker credits winners or retains losing stakes.
All cash-flow and record-keeping occur on the bookmaker’s central ledger.
\end{enumerate}
\end{minipage}\hfill
% -------- RIGHT COLUMN -------------------------------------------------
\begin{minipage}[t]{0.40\textwidth}\vspace{0pt}
\centering
\begin{tikzpicture}[
node distance = 1cm, % tighter vertical gap
every node/.style = {
rectangle, draw=gray!60, fill=gray!12,
rounded corners=2pt,
text width=4.5cm, align=left, font=\scriptsize
}
]
\node (post) {\textbf{Odds Posting}};
\node (bet) [below=of post] {\textbf{Bet Placement}};
\node (accept) [below=of bet] {\textbf{Instant Acceptance}};
\node (stagn) [below=of accept] {\textbf{Market Stagnation}};
\node (settle) [below=of stagn] {\textbf{Settlement}};
\foreach \i/\j in {post/bet, bet/accept, accept/stagn, stagn/settle}
\draw[-{Stealth[length=1.5mm]}] (\i) -- (\j);
\end{tikzpicture}
\end{minipage}
\end{center}
\begin{tcolorbox}[title={Evaluation Metrics — Traditional Bookmaker Systems},
colback=gray!3, colframe=gray!60!black,
boxrule=0.6pt, arc=3pt,
left=6pt, right=6pt, top=4pt, bottom=4pt,
before skip=10pt, after skip=10pt]
\begin{enumerate}[label=\arabic*., leftmargin=*, itemsep=6pt]
\item \textbf{Protocol Versatility}~\failBox{\ding{55}}\\
\emph{Performance}: Limited for long-tail events—the bookmaker lists only markets it chooses to underwrite.\\
\emph{\color{red} Key Shortcoming}: Niche or bespoke events may be unavailable.
\item \textbf{System Acquisition Capability}~\failBox{\ding{55}}\\
\emph{Performance}: Static odds update slowly and incorporate information asymmetrically.\\
\emph{\color{red} Key Shortcoming}: Prices lag real-time sentiment; informed bettors can exploit stale lines.
\item \textbf{Price Sum-to-One }~\failBox{\ding{55}}\\
\emph{Performance}: Holds by construction (posted odds sum to $>\!1$ to embed margin).\\
\emph{\color{red} Key Shortcoming}: Over-round introduces systematic bias away from true probabilities.
\item \textbf{Automatic Settlement}~\failBox{\ding{55}}\\
\emph{Performance}: Settlement relies on a central party that must honor payouts.\\
\emph{\color{red} Key Shortcoming}: Requires trust in bookmaker solvency—single point of failure.
\end{enumerate}
\end{tcolorbox}
\subsection{CLOB + CTF}
\begin{tcolorbox}[
title={Mechanism Overview },
enhanced,
colback=white,
colframe=blue!80!black,
colbacktitle=blue!10,
fonttitle=\bfseries\sffamily,
coltitle=black,
borderline west={2.5pt}{0pt}{blue!80!black}, % ← vertical stripe
boxrule=0.6pt, arc=2pt,
left=8pt,right=6pt,top=4pt,bottom=4pt]
The most popular architecture for prediction markets today is the Central Limit Order Book (CLOB) + Conditional Token Framework (CTF) combination, already used by leading platforms such as Polymarket and Kalshi.
\begin{itemize}
\item \textbf{CLOB:} an on-chain limit-order book in which each tradable asset has its own order book. Users can submit limit or market orders and have them matched freely.
\item \textbf{CTF:} A mechanism for issuing conditional tokens. It mints (or burns) mutually exclusive asset pairs—such as YES/NO contracts—when the specified event conditions are met.
\end{itemize}
\end{tcolorbox}
%-----------------------------------
\begin{tcolorbox}[sharp corners,
sidebyside,
sidebyside gap=8mm,
valign=center,colback=gray!0,
colframe=gray!60!black]
\textbf{Initializing Multiple-choice Option}
In a multi-outcome prediction market with mutually exclusive results
(A, B, C), the mechanism creates a separate \emph{binary} sub-market
for each outcome.\\
\textbf{For simplicity,} we only use N = 3 for illustration.
\tcblower % --- everything after this goes to the right half ---
\begin{tabular}{@{}lccc@{}}
\toprule
\textbf{Outcome} & A & B & C \\
Yes Token & A-Yes & B-Yes & C-Yes \\
No Token & A-No & B-No & C-No \\
\bottomrule
\end{tabular}
\end{tcolorbox}
\subsubsection*{Detailed order matching Mechanism: two parallel order matching path}
%-----------------------------------
\begin{tcolorbox}[sharp corners,
sidebyside,
sidebyside gap=8mm,
valign=center,colback=gray!0,
colframe=gray!60!black]
\textbf{\color{red} CLOB matching (vertical)}
\begin{itemize}
\item In each sub-market (e.g., A-Yes / A-No):
\item Users may place buy or sell orders on either the YES or NO asset.
\item If bid and ask prices cross (e.g., bid = 0.7, ask = 0.7), the system matches the order.
\item The process is almost identical to a conventional exchange’s order book.
\end{itemize}
\tcblower % --- everything after this goes to the right half ---
% ------- two slightly smaller boxes side-by-side (quotes no longer overlap) -------
\begin{tikzpicture}[x=1cm,y=1cm,font=\small]
% ===== shared dimensions =================================================
\def\W{3.8}
\def\H{4.6}
\def\Mid{2.3}
\def\Gap{4cm}
% =========================================================================
% ---------- FIRST BOX : YES ---------------------------------------------
\draw[thick,rounded corners=8pt] (0,0) rectangle (\W,\H);
\draw[thick] (0,\Mid) -- (\W,\Mid);
\node at (\W/2,\H-0.4) {\textbf{YES}};
% Bid / Ask labels
\node[anchor=west] at (0.1,\Mid+0.45) {Bid};
\node[anchor=west] at (0.1,\Mid-0.45) {Ask};
% red CLOB line
\draw[red,line width=2pt] (\W/2,\Mid+0.5) -- (\W/2,\Mid-0.5);
\node[anchor=west] at (\W/2,\Mid+0.15) {\textbf{\color{red} CLOB}};
% Bid quotes (shifted down 0.2 cm)
\foreach \y/\l in {3.8/0.72, 3.4/0.71, 3.0/0.70}{
\node at (\W/2,\y) {\l};
}
% Ask quotes (unchanged)
\foreach \y/\l in {1.4/0.70, 1.0/0.69, 0.6/0.68}{
\node at (\W/2,\y) {\l};
}
% ---------- SECOND BOX : NO ---------------------------------------------
\begin{scope}[xshift=\Gap]
\draw[thick,rounded corners=8pt] (0,0) rectangle (\W,\H);
\draw[thick] (0,\Mid) -- (\W,\Mid);
\node at (\W/2,\H-0.4) {\textbf{NO}};
% Bid / Ask labels
\node[anchor=west] at (0.1,\Mid+0.45) {Bid};
\node[anchor=west] at (0.1,\Mid-0.45) {Ask};
% red CLOB line
\draw[red,line width=2pt] (\W/2,\Mid+0.5) -- (\W/2,\Mid-0.5);
\node[anchor=west] at (\W/2,\Mid+0.15) {\textbf{\color{red} CLOB}};
% Bid quotes (shifted down 0.2 cm)
\foreach \y/\l in {3.8/0.32, 3.4/0.31, 3.0/0.30}{
\node at (\W/2,\y) {\l};
}
% Ask quotes (unchanged)
\foreach \y/\l in {1.4/0.29, 1.0/0.28, 0.6/0.27}{
\node at (\W/2,\y) {\l};
}
\end{scope}
\end{tikzpicture}
\end{tcolorbox}
%-----------------------------------
\begin{tcolorbox}[sharp corners,
sidebyside,
sidebyside gap=8mm,
valign=center,colback=gray!0,
colframe=gray!60!black
]
\textbf{\color{green!70!black} CTF (horizontal)}
\begin{itemize}
\item A unique method introduced by the CTF.
\item When the system detects that the sum of bid prices for an option’s Yes and No assets equals 1 (e.g.\ Yes bid 0.6 + No bid 0.4 = 1), it collects 1 USDC and mints a corresponding YES/NO token pair via the CTF, delivering the tokens to the trader.
\item Similarly, when the sum of ask prices for an option’s YES and NO assets equals 1, the system burns the corresponding Yes/No token pair and refunds 1 USDC to the user.
\item This internal auto-mint/redemption mechanism lets trades complete automatically when no counterparty is available.
\end{itemize}
\tcblower % --- everything after this goes to the right half ---
\begin{tikzpicture}[scale=1.6,transform shape,x=1cm,y=1cm,font=\small]
% ===== shared dimensions =========================================
\def\W{4.6} % main box width
\def\H{1.8} % main box height
\def\Mid{0.9} % main box vertical centre
\def\HouseW{1.6}% house body width
\def\HouseH{0.9}% house body height
% =================================================================
% ---- main YES/NO box --------------------------------------------
\draw[thick,rounded corners=6pt] (0,0) rectangle (\W,\H);
\draw[thick] (\W/2,0) -- (\W/2,\H);
% headers
\node at (\W/4,\H-0.25) {\textbf{YES}};
\node at (3*\W/4,\H-0.25) {\textbf{NO}};
% green CFT equilibrium bar
\draw[green!70!black,line width=2pt] (1.5,\Mid) -- (\W-1.5,\Mid);
% prices
\node at (\W/4,\Mid) {0.60};
\node at (3*\W/4,\Mid) {0.40};
% ---- vertex where the two legs meet -----------------------------
\coordinate (v) at (\W/2, -0.3);
% ---- CFT House ---------------------------------------------------
\coordinate (houseBL) at ({\W/2-\HouseW/2}, -2.0);
\coordinate (houseBR) at ($(houseBL) + (\HouseW,0)$);
\coordinate (houseTL) at ($(houseBL) + (0,\HouseH)$);
\coordinate (houseTR) at ($(houseBR) + (0,\HouseH)$);
\coordinate (roofPeak) at ({\W/2}, {-2.0+\HouseH+0.5}); % roof apex
\draw[thick,fill=gray!10] (houseBL) rectangle (houseTR); % body
\draw[thick,fill=gray!20] (houseTL) -- (roofPeak) -- (houseTR) -- cycle; % roof
\node[font=\scriptsize\bfseries] at ($(houseBL)!0.5!(houseTR)$) {{\color{green!70!black} CTF:} mint / burn token pair};
% ---- arrows ------------------------------------------------------
% two skewed legs merging at the vertex
\draw[thick,<->] (1.1,0) -- (v);
\draw[thick,<->] (3.5,0) -- (v);
% vertical arrow from vertex straight down to the CFT House
\draw[thick,->] (v) -- (roofPeak);
\end{tikzpicture}
\end{tcolorbox}
%-----------------------------------
\begin{tcolorbox}[sharp corners,
sidebyside,
sidebyside gap=8mm,
valign=center,colback=gray!0,
colframe=gray!60!black]
\textbf{\color{red} CLOB (vertical)} + \textbf{\color{green!70!black} CTF (horizontal)}
\begin{itemize}
\item Each of the N possible outcomes is turned into N independent binary prediction sub-markets.
\item Every sub-market contains a pair of YES/NO assets that trade via a vertical order book of limit orders.
\item The CTF layer introduces automatic horizontal combo trades: when the summed prices of YES + NO hit a specified condition, the system can mint or burn the corresponding liquidity pair.
\item By combining vertical limit-order posting with horizontal conditional matching, the system maximizes liquidity utilization and enhances price discovery.
\end{itemize}
\tcblower % --- everything after this goes to the right half ---
% ---------- RIGHT PANEL : updated TikZ ----------
\resizebox{.8\linewidth}{!}{%
\begin{tikzpicture}[x=1cm,y=1cm,font=\small]
% size helpers
\def\W{8} % width
\def\H{10} % height
\def\Mid{5} % horizontal mid-line (Bid↔Ask)
% --- outer rounded frame ---
\draw[thick,rounded corners=8pt] (0,0) rectangle (\W,\H);
% --- grid (kept in black) ---
\draw[thick] (\W/2,0) -- (\W/2,\H); % YES / NO split
\draw[thick] (0,\Mid) -- (\W,\Mid); % Bid / Ask split
% ===== coloured emphasis strokes =====
% (1) vertical – CLOB, drawn in RED and long enough to cross the Bid/Ask line
\draw[red, line width=2 pt] (2,\Mid+1.3) -- (2,\Mid-1.3);
\node[anchor=west] at (2,\Mid+0.25) {\textbf{\color{red} CLOB}};
% (2) horizontal – CTF, drawn in GREEN and long enough to cross the YES/NO divide
\draw[green!70!black, line width=3pt] (2.5,6.7) -- (\W-2.5,6.7);
\node[anchor=west] at (\W/2,6.9) {\textbf{\color{green!70!black} CTF}};
% ===== column headers =====
\node at (\W/4,\H-0.6) {\textbf{YES}};
\node at (3*\W/4,\H-0.6){\textbf{NO}};
% ===== side labels =====
\node[anchor=east] at (-0.3,0.75*\H) {Bid};
\node[anchor=east] at (-0.3,0.25*\H) {Ask};
% ===== Bid quotes =====
\foreach \y/\l in {8.5/0.72, 7.6/0.71, 6.7/0.70}{
\node at (\W/4,\y) {\l};
}
\foreach \y/\l in {8.5/0.32, 7.6/0.31, 6.7/0.30}{
\node at (3*\W/4,\y) {\l};
}
% ===== Ask quotes =====
\foreach \y/\l in {3.3/0.70, 2.4/0.69, 1.5/0.68}{
\node at (\W/4,\y) {\l};
}
\foreach \y/\l in {3.3/0.29, 2.4/0.28, 1.5/0.27}{
\node at (3*\W/4,\y) {\l};
}
\end{tikzpicture}}
\end{tcolorbox}
\begin{tcolorbox}[title={Evaluation Metrics — CLOB + CTF },
colback=gray!0, colframe=gray!60!black,
boxrule=0.6pt, arc=3pt,
left=6pt, right=6pt, top=4pt, bottom=4pt,
before skip=10pt, after skip=10pt]
\begin{enumerate}[label=\arabic*., leftmargin=*, itemsep=6pt]
\item \textbf{Protocol Versatility}~\failBox{\ding{55}}\\
\emph{Performance:} This model offers exceptionally high trading efficiency and an excellent user experience because it closely mirrors traditional order-book markets, making it easy for users to understand and adopt.\\
\emph{\color{red}Key Shortcoming:} Owing to the CTF design, which maps each of the $N$ outcomes to $2N$ tokens and thus disperses liquidity, niche or bespoke events may lack sufficient market depth.
\item \textbf{System Acquisition Capability}~~\failBox{\ding{55}}\\
\emph{\color{red}Key Shortcoming:} Because of the order-matching mechanism, price quotations may not respond sensitively to changes in market beliefs and therefore may fail to reflect the collective belief about an event.
\item \textbf{Price Sum-to-One }~\successBox{\ding{51}}\\
\emph{Performance:} Atomic swaps ensure that the prices sum to one.
\item \textbf{Smooth Settlement}~\successBox{\ding{51}}\\
\end{enumerate}
\end{tcolorbox}
\subsection{Robin Hanson's LMSR}
% ----------- boxed “Mechanism Overview” -----------------
\begin{tcolorbox}[
title={Mechanism Overview },
colback=white,
colframe=blue!80!black,
colbacktitle=blue!10,
fonttitle=\bfseries\sffamily,
coltitle=black,
borderline west={2.5pt}{0pt}{blue!80!black},
boxrule=0.6pt, arc=2pt,left=8pt,right=6pt,top=4pt,bottom=4pt]
Robin Hanson's LMSR, or Logarithmic Market Scoring Rule, is a market scoring rule he invented to facilitate prediction markets. LMSR is designed to provide continuous liquidity in these markets, ensuring that participants can always buy or sell shares without the market becoming illiquid. It works by adjusting prices logarithmically based on the quantity of shares bought or sold, which helps aggregate information efficiently and incentivizes accurate forecasting.
\end{tcolorbox}
\begin{figure}[h!]
\centering
% ==================== LEFT MINIPAGE (formulas) ============================
\begin{minipage}[t]{0.46\textwidth}\vspace{0pt} % ← adjust width if needed
\begin{tcolorbox}[title={Mathematical representation for LMSR},
colback=gray!0, colframe=gray!60!black,
boxrule=0.6pt, arc=3pt,
left=6pt, right=6pt, top=4pt, bottom=4pt,
before skip=10pt, after skip=10pt]
\begin{enumerate}[label=\arabic*., leftmargin=*, itemsep=6pt]
\item \textbf{Reserve-Supply Relation}~\[ r_U = b \cdot \left(
e^{\frac{s_Y}{b}} + e^{\frac{s_N}{b}}
\right) \]
\item \textbf{Price of Token Yes}~ \[ p_{YU} = \frac{\partial r_U}{\partial s_Y} =\frac{e^{\frac{s_Y}{b}}}
{e^{\frac{s_Y}{b}} + e^{\frac{s_N}{b}}} \]
\item \textbf{Price of Token No}~ \[ p_{NU} =\frac{\partial r_U}{\partial s_N}=\frac{e^{\frac{s_N}{b}}}
{e^{\frac{s_Y}{b}} + e^{\frac{s_N}{b}}} \]
\end{enumerate}
For simplicity, we only use N = 2 for illustration.
\end{tcolorbox}
\end{minipage}\hfill
% ==================== RIGHT MINIPAGE (diagram) ============================
\begin{minipage}[t]{0.50\textwidth}\vspace{0pt}
\centering
\begin{tikzpicture}
\begin{axis}[
xlabel={$s_Y$ (token YES supply)},
ylabel={$s_N$ (token NO supply)},
zlabel={$r_U$ reserve},
domain=0:5, samples=26,
view={60}{28},
colormap/viridis,
mesh/ordering=y varies,
zmin=0,
grid=both, minor tick num=1
]
% -------- surface: r_U = b * ln(e^{x}+e^{y}) -----------------------
\addplot3[surf, z buffer=sort, opacity=0.85] {ln(exp(x)+exp(y))};
\end{axis}
\end{tikzpicture}
\end{minipage}
\caption{(Left) Cost function and marginal prices for a two-outcome LSMR.\\
\hspace{2cm}%
(Right) Corresponding cost surface \(r_U=b \cdot \bigl(e^{s_Y/b}+e^{s_N/b}\bigr)\) with \(b=1\).}
\label{fig: lsmr-cost-surface}
\end{figure}
\begin{tcolorbox}[title={Evaluation Metrics — LMSR},
colback=gray!5, colframe=gray!60!black,
boxrule=0.6pt, arc=3pt,
left=6pt, right=6pt, top=4pt, bottom=4pt,
before skip=10pt, after skip=10pt]
\begin{enumerate}[label=\arabic*., leftmargin=*, itemsep=6pt]
\item \textbf{Protocol Versatility}~\successBox{\ding{51}}
\item \textbf{System Acquisition Capability}~\failBox{\ding{55}}\\
\emph{Performance}: Supposing b = 1, $s_Y$ = $s_N$ = 0, then: $ p_{YU} = p_{NU} = \frac{1}{2}.$\\
\emph{\color{red} Key Shortcoming:} The initial price for ”Yes” and "NO" is always 0.5; not catching the market belief when setup.
\item \textbf{Probability Equals One}~\successBox{\ding{51}}\\
\emph{Performance}: $ p_{YU} + p_{NU} = 1. $
\item \textbf{Smooth Settlement}~\failBox{\ding{55}}
\emph{Performance}: Supposing \( b = 1 \), \( s_Y = 10 \), and \( s_N = 100 \), then: $ r_U = b \cdot \ln\left( e^{10} + e^{100} \right) \approx 100.$ \\
\emph{\color{red} Key Shortcoming:} The liquidation price for ”Yes” would be 10 and for ”No” it would be 1; resulting in an clumsy and skewed settlement process.
\end{enumerate}
\end{tcolorbox}
\section{The Proposed BAPMM}
To overcome the weakness of existing Liquidity Mechanisms (Order Matching) in prediction market, we will contrusct our protocol from first principle and showing the designed BAPMM pass through all Evaluation Metrics in Prediction Market.
\subsection{Static Model}
\begin{tcolorbox}[title = First Principle, colback=gray!0,
colframe=gray!60!black ]
\begin{enumerate}[label=\arabic*., leftmargin=*, itemsep=3pt]
\item \textbf{Boundary Condition: Ensuring One-to-One Final Settlement } \\
\emph{Insight:}
A crucial design goal in prediction markets is that upon outcome revelation, exactly one token (the winning outcome)
redeems for 1 USD each, while other outcome tokens become worthless. To achieve this in an automatic fashion (i.e.,
no extra redistribution steps), the supplies of all tokens must match the USD reserve, resulting: \[ s_1 = \ldots =s_i \ldots = s_N =r_U \]
\item \textbf{Construction:}
In the context of prediction markets, we can consider a mint-and-burn mechanism for a \emph{token set}\footnote{This pertains to a token set mapping rather than a total differential, so notation like \(\,dr_U = ds_{outcome_1} + \cdots + ds_{outcome_N}\)\, is unclear.} that defines the relationship between the USD token reserve and the supplies of each outcome token via a \textit{mapping function}:
\[
\hat{T} : r_U \longrightarrow\; S,
\qquad
\hat T(x)=x\,\mathbf 1_{N},
\]
\noindent Conversely, the inverse mapping function \( \hat{T}^{-1}\) translates the USD reserve back into the required outcome-token supplies
\end{enumerate}
\end{tcolorbox}
\subsubsection*{Illustrative Example}
% -------- LEFT COLUMN -------------------------------------------------
\begin{minipage}[t]{0.54\textwidth}\vspace{0pt}
\begin{itemize}[leftmargin=*, itemsep=4pt, topsep=0pt]
\item \textbf{Mapping Function and Inverse.}
Suppose there are three possible outcomes: Yes (Y), No (N), and Draw (D).
\[
\hat{T}(r_U)= \{s_Y, s_N, s_D\},\qquad r_U = \hat{T} \bigl(\{s_Y, s_N, s_D\}\bigr).
\]
\item \textbf{Current State.}
If the current supply is 10 for each outcome,
\[
\hat{T}(10)= \{10, 10, 10\},\qquad 10 = \hat{T} \bigl(\{10, 10, 10\}\bigr).
\]
\item \textbf{\color{green!70!black}{Deposit (Mint)}.}
A user pays \$1 USD and receives the set \(\{1,1,1\}\).
The USD reserve rises to 11; total supply becomes \(\{11,11,11\}\).
\item \textbf{\textcolor{red}{Withdraw (Burn)}.}
To redeem \$1 USD the user must return \emph{all} three tokens.
Afterward the reserve is 9 and supply is \(\{9,9,9\}\).
\end{itemize}
\end{minipage}\hfill
% -------- RIGHT COLUMN -------------------------------------------------
\begin{minipage}[t]{0.5\textwidth}\vspace{0pt}
\centering
\includegraphics[width=\linewidth]{Token_Set.png}
% If you load \usepackage{caption} in the preamble:
\captionof{figure}{Illustration of the Mint-and-Burn Mechanism}
\end{minipage}
\begin{tcolorbox}[title={ Why This Matters} ]
\begin{itemize}
\item If outcome~\(i\) is prevailed, its supply \(s_{outcome_i}\) \textit{alone} equals \(r_U\). Hence each token of outcome~\(i\) redeems for exactly 1~USD, \emph{automatically} ensuring a one-to-one liquidation price.
\item No complex fractionation or reallocation is needed at settlement: once the real-world event is known, holders of the winning token can directly claim their 1~USD redemption from the bonding curve, matching token supply to the entire USD reserve.
\item This design cements a clear equivalence: "\(s_{outcome_i} = r_U\)" means the entire USD reserve backs the single winning token supply, seamlessly facilitating final payout.
\end{itemize}
\end{tcolorbox}
\subsection{Model with Uniswap-like LP}
We can embedding a crowd swap mechanism into the static model. The bonding is agnostic the existence of external swap mechanism and vice versa.
\begin{tcolorbox}[title={User Side},
colback=gray!0,
colframe=gray!60!black]
\begin{enumerate}[label=\arabic*., leftmargin=*, itemsep=6pt]
\item \textbf{Deposit into the Bonding Curve:}~\\ We deposit \(\Delta\) USD tokens into the bonding curve, receiving token set \(\Delta\)$ S $ in return.
\item \textbf{Keep the token i ans throw the rest into the pool}~
\[
\prod_{i=1}^N r_{i} ^{w_i} = k
\]
\item \textbf{Total Received Amount of Token i}~ \\
Including the initial \(\Delta\) token i minted from the bonding curve, the total amount of token i is:
\begin{align}
& \Delta +r_i(1-\prod_{j\neq i} r_j^{\frac{w_j}{w_i}}(r_j+\Delta)^{-\frac{w_j}{w_i}}) \\
= & \Delta +r_i(1-\prod_{j\neq i} (1- \frac{w_j}{w_i} \cdot r_j^{-1} \cdot \Delta+ \ldots)
\end{align}
\end{enumerate}
\textbf{ps1:} Withdraw (Burn) is the reversed process of aforementioned Deposit (Mint) process, we omit the details.
\end{tcolorbox}
\begin{figure}[h!]
\centering
\includegraphics[width=1\textwidth]{BAPMM_architecture.png}
\caption{Overall and Individual State }
\end{figure}
\begin{tcolorbox}[title={Liquidity Provider: },
colback=gray!0, colframe=gray!60!black,
boxrule=0.6pt, arc=3pt,
left=6pt, right=6pt, top=4pt, bottom=4pt,
before skip=10pt, after skip=10pt]
\begin{enumerate}[label=\arabic*., leftmargin=*, itemsep=6pt]
\item \textbf{Deposit into the Bonding Curve:}~\\ We deposit \(\Delta\) USD tokens into the bonding curve, receiving token set S in return.
\item \textbf{Increasing Liquidity:}~\\
Throw token set S to Liquidity Pool, and receiving LP
\item \textbf{Receiving lp token} ~
$lp_{diff} = \prod_{i=1}^N (r_i+\Delta)^{w_i} -\prod_{i=1}^N (r_i)^{w_i} $
\end{enumerate}
\textbf{ps1:} Removing is the reversed process of Increasing Liquidity, we omit the details.
\end{tcolorbox}
\begin{tcolorbox}[title={Result Settlement},
colback=gray!0, colframe=gray!60!black,
boxrule=0.6pt, arc=3pt,
left=6pt, right=6pt, top=4pt, bottom=4pt,
before skip=10pt, after skip=10pt]
\begin{enumerate}[label=\arabic*., leftmargin=*, itemsep=6pt]
\item \textbf{Oracle} Because a blockchain has no inherent knowledge of real-world events, an external oracle must provide definitive outcome data for settlement. Among possible solutions, a zero-knowledge trustless (ZK-TLS) approach or a community-driven voting mechanism represent two major design choices.
\item \textbf{When} the result of a certain $outcome_i$ prevailed and updated within the bonding curve contract, all the holder of this specific outcome can redeem 1 USD per $outcome_i$ token.
\item \textbf{Now} the bonding contract only takes $\{0, \cdots, \Delta outcome_i, ...., 0 \} $ for input.
\end{enumerate}
\end{tcolorbox}
\subsection{BAPMM Supremacy}
\begin{tcolorbox}[title=Sum-to-One Verification, colback=gray!1, colframe=gray!60!black]
Given the relative price definition
\[
p_{iU}\;=\;
\frac{1}{\displaystyle\sum_{j}\frac{w_j}{w_i}\,\frac{r_i}{r_j}},
\]
pick an arbitrary reference index \(k\) and multiply every numerator–denominator pair by \(w_k r_k\):
\[
\frac{w_j}{w_i}\,\frac{r_i}{r_j}
=\frac{w_j r_k}{w_k r_j}\;.
\]
Hence
\[
\sum_{i}p_{iU}
=\sum_{i}\frac{w_k r_k}{\displaystyle\sum_{j}w_j r_k}
=\frac{\sum_{i}w_k r_k}{\sum_{j}w_j r_k}
=1.
\]
Therefore the prices are properly normalised: \(\sum_{i}p_{iU}=1\).
\end{tcolorbox}
\begin{tcolorbox}[title=An Art of Subtraction,
colback=gray!0,
colframe=gray!60!black]
Philosophically speaking, BAPMM presents an art of subtraction compared with CLOB
See \href{https://docs.polymarket.com/#create-and-place-an-order}{polymarket doc} for more details). In one word, the CLOB first match the price in bulk, then mint or burn the token pair; whilst for us, we mint or burn the token pair first and then match the price algorithmically. They converge asymptotically when the trading volume tends to infinity, similar to the comparison between Uniswap-like protocols and order books \footnote{Uniswap-like liquidity providers are essentially promised to sell or buy each token from a pair within a certain price range as a market maker. Also, any trade on Uniswap-like protocols will affect the price, unlike the buck trade in order book.}. Still, our protocol further removes the role of the order operator in CLOB and enable an continuous algorithmic pricing.
\end{tcolorbox}
\begin{table}[H]
\centering
\renewcommand{\arraystretch}{2}
\begin{tabular}{lcccc}
% \toprule
& \textbf{Versatility} & \textbf{Acquisition} & \textbf{$\sum p_i=1$} & \textbf{Settlement}\\
\midrule
\mechanism{Traditional Bookmaker} & ~\failBox{\ding{55}} & ~\failBox{\ding{55}} & ~\failBox{\ding{55}} & ~\failBox{\ding{55}} \\
\mechanism{ CLOB+ CTF } & ~\failBox{\ding{55}} & ~\failBox{\ding{55}} & ~\successBox{\ding{51}} & ~\successBox{\ding{51}}\\
\mechanism{LMSR } & ~\successBox{\ding{51}} & ~\failBox{\ding{55}} & ~\successBox{\ding{51}} & ~\failBox{\ding{55}} \\
\mechanism{BAPMM } & ~\successBox{\ding{51}} & ~\successBox{\ding{51}}& ~\successBox{\ding{51}} & ~\successBox{\ding{51}} \\
% \bottomrule
\end{tabular}
% \caption{Scorecard of existing mechanisms against the four metrics.}
\label{tab:mechanism-scorecard}
\end{table}
% ----------------------------------------
\begin{tcolorbox}[title= BAPMM's performance ]
\begin{itemize}
\item \textbf{Direct Outcome Pricing}: The bonding curve can automatically adjust prices based on overall supply and demand, ensuring that even low-probability outcomes have meaningful pricing without relying on active order matching.
\item \textbf{Long-tail compatible }: In scenarios where the CLOB fails to match orders due to imbalanced prices, the bonding curve can still facilitate transactions by adjusting the token supply according to the algorithm, ensuring that the market remains functional and accurately represents collective probabilities.
\item \textbf{Dispersed knowledge} is traded in a single marketplace, and prices dynamically encode the crowd’s evolving beliefs about each event, enabling the system to capture broad signals and update those beliefs in real time.
\item \textbf{Only N token} is needed for outcome representation.
\end{itemize}
\end{tcolorbox}
\begin{tcolorbox}[title=Price–probability Duality]
\textbf{Key Insight: Internal Bonding Preserves the Sum}
\emph{No matter how users exchange tokens} among themselves (Y, N for D, or vice versa), the \textbf{overall system supply} for each outcome is still governed by the mint-and-burn mechanism. The external market merely re-distributes existing Y, N and D among participants; it does not create or destroy any additional outcome tokens. This yields a notable \emph{duality} between \emph{price} and \emph{probability}:
\begin{itemize}
\item \textbf{Local/Derivative Angle:}
Each incremental trade can be broken down via partial derivatives, showing the cost for "a bit more Y" in place of "a bit less N and D". Summing these local prices always yields a full "1 unit of value".
\item \textbf{Global/Fraction Angle:}
Globally, \(Y\), \(N\),\(D\) still partition a single "sample space", so \(Y\text{'s fraction} + N\text{'s fraction}+D\text{'s fraction} = 1\). By maintaining a one-to-one match between token supply and reserve at \emph{all} times, the system mirrors a probability-like partition of outcomes, guaranteeing \(\sum \text{(probability)} = 1\).
\item \textbf{Protocol Neutrality vs.\ Market Beliefs:}
Internally, the bonding curve itself is "behind ignorance wall", initially treating both outcomes as equally likely (\(p= \tfrac{1}{3}\) each, or \(\tfrac{1}{N}\) in the \(N\)-outcome case). In quantum physics language,
the bonding curve maintains the outcome set a as a superposition within a box (\( \ket{u}=\frac{1}{\sqrt{3}}( \ket{1}+\ket{2}+\ket{3})\) and collapse to one result when settlement. However, once outcome tokens are free to trade on external markets, their prices adjust based on participants’ knowledge and incentives, reflecting the evolving \emph{market-driven} estimate of each outcome’s probability.
\item \textbf{What Restriction Do We Impose?}
In an efficient market, the relative prices satisfy
\[
\frac{p_{iU}}{p_{jU}} = p_{ij},
\]
However, \( \sum_{i=1}^N p_{iU} \) can freely vary and is not fixed by this ratio condition alone; therefore, our key design choice is to \emph{restrict} that sum to be exactly one , effectively forcing
\[\sum_{i=1}^N p_{iU} = 1 \]
and thereby enforcing a "probability-like" partition of value across the N outcomes.
\item \textbf{For simplicity,} we only use N = 3 for illustration.
\end{itemize}
\end{tcolorbox}
\section{Open Questions} Below, we highlight potential challenges and opportunities for broader applications and future research.
\begin{tcolorbox}[title = Abnormal Situations, colback=gray!0]
Consider a scenario in which the oracle fails or the real‑world outcome is indeterminate—for example, the event never unfolds clearly, or perhaps not at all. When the oracle produces an incorrect answer or cannot reach a decision, the crucial question is how, and by whom, a fallback procedure should be triggered. Extreme edge cases—such as catastrophic blockchain failure or other existential events—demand contingency mechanisms that can override or amend the original settlement rules.
\end{tcolorbox}
\begin{tcolorbox}[title=Integration with Other DeFi,
colback=gray!0,
colframe=gray!60!black]
Because each outcome token is minted as a standard fungible asset, users can freely incorporate these tokens into any compatible DeFi protocol. For instance, they may provide liquidity on Uniswap, stake on Governance or lend tokens on Aave, or otherwise leverage their holdings in various yield strategies. This flexibility ensures outcome tokens retain the composability and portability characteristic of other widely adopted ERC-20 (or similar) tokens in the DeFi ecosystem.
\end{tcolorbox}
\begin{tcolorbox}[title=Governance, colback=gray!0]
While the protocol automates liquidity management, it still requires a governance layer for upgrades, policy decisions, and emergency measures. True decentralization calls for community-based discussions and voting, so no single entity can unilaterally impose changes. Moreover, unknown unknowns—unforeseen issues beyond the scope of standard fail-safes—may demand human intervention or protocol alterations. Robust governance structures help maintain the system’s integrity when confronting such exceptional challenges.
\end{tcolorbox}
\begin{tcolorbox}[title =Mapping Event Probability Tokenization to Options and Futures, colback=gray!0]
This broader scope could integrate price discovery, hedging, and settlement across a range of traditional and novel derivatives markets, all within a composable, on-chain framework.
\end{tcolorbox}
\begin{tcolorbox}[title = Causal Inference, colback=gray!0]
The process to capture the structure of crowd‘s information exchange might enhance the capability of causal inference beyond the current LLM
\end{tcolorbox}
\section{Acknowledgement}
We gratefully acknowledge the joint efforts of catslovefish.eth, who developed the theoretical foundations of the price–probability duality and served as the manuscript’s principal typist; Edward Lee, whose insightful direction shaped the project’s scope and focus; and Arwen Yi, who refined the paper structure enhancing readability. Cat also extends personal thanks to two former physics supervisors, whose mentorship fostered both a deep appreciation of many-body physics and a lasting joy in exploring elegant theoretical models, as well as to a media artist who introduced me to the film Solaris, offering a fresh emotional perspective on “planetary wisdom.”
\newpage
\end{document}